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We investigate the properties of the dispersion tensor in porous media by making use of the method of homogenization by multiple scale expansions, which gives the dispersion model from the pore scale description. Results are valid for random or periodic structures as well. By investigating the domain of relatively small Péclet numbers, we show that the dispersion tensor is asymmetric in some cases. We also demonstrate that in the whole range of validity of the dispersion equation, the dispersion tensor verifies D dis i j (grad X p) = D dis ji (−grad X p). Links with Onsager's relations are given. Numerical investigations at finite Péclet numbers confirm the possible asymmetry of the dispersion tensor.

This paper describes a new formulation of the vapor−liquid equilibrium problem. The proposed model reveals a great mathematical similarity between the stability test problem and the isothermal flash problem, particularly in regions of interest in reservoir engineering near phase boundaries and in the vicinity of critical points. This similarity is mathematically studied and then explored to find a means of minimizing the Gibbs free energy in a more efficient way. The Peng−Robinson cubic equation of state was used to represent the phase behavior of multicomponent mixtures. The simulated annealing (SA) algorithm was used to carry out isothermal flash calculations and phase stability tests. The SA algorithm is essentially an iterative random search procedure whose moves are adapted by some suitable mechanism. It is robust and easy to implement. The algorithm was tested for several hydrocarbon mixtures. The few experimental data available in the literature were compared with theoretical predictions. The proposed methodology produced high-quality solutions.

The spatial variations in porous media (aquifers and petroleum reservoirs) occur at all length scales (from the pore to the reservoir scale) and are incorporated into the governing equations for multiphase flow problems on the basis of random fields (geostatistical models). As a consequence, the velocity field is a random function of space. The randomness of the velocity field gives rise to a mixing region between fluids, which can be characterized by a mixing length = (t). Here we focus on the scale-up problem for tracer flows. Under very general conditions, in the limit of small heterogeneity strengths it has been derived by perturbation theories that the scaling behavior of the mixing region is related to the scaling properties of the self-similar (or fractal) geological heterogeneity through the scaling law (t) ∼ t γ , where γ = max{1/2, 1 − β/2}; β is the scaling exponent that controls the relative importance of short vs. large scales in the geology. The goals of this work are the following: (i) The derivation of a new, mathematically rigorous scaling analysis for the tracer flow problem subject to self-similar heterogeneities. This theoretical development relates the large strength to the small strength heterogeneity regime by a simple scaling of solutions. It follows from this analysis that the scaling law derived by perturbation theory is valid for any strength of the underlying geology, thereby extending the current available results. To the best of the knowledge of the authors this is the only rigorous result available in the literature for the large strength heterogeneity regime. (ii) The presentation of a Monte Carlo study of highly resolved simulations, which are in excellent agreement with the predictions of our new theory. This indicates that our Monte Carlo results are accurate and can be applied to other models for stochastic geology.

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